YES they will. But if you can't prove they sold it to you, good luck.
About six months ago I broke a socket I bought through a former employer through them. The first time I contacted Grainger (email, don't have a Grainger rep anymore) I ran into a brick wall -- guy was no help, gave no answers at all, wouldn't even send me a form letter about warranty policy.
Then about a month ago I tried again, what the hell I thought. I'd already replaced the socket with a Mac. I got in touch (email again) with a woman who knew the deal and was eager to make the transaction positive. All she needed from me, apart from the item number, was my former company's name and a ballpark purchase date. She found the sales record, and less than a week after first contacting her I got a replacement in the mail. Didn't even have to send in the broken one.
For what it's worth, she did phrase her reply as though it wasn't a problem when the necessary warranty item is a relatively inexpensive one. I'm imagining if I had a couple-hundred dollar, 70 pound item I'd probably have to go to a Grainger brick and mortar.
